Abstract
We study different algebraic and algorithmic constructions related to the scalar product on the space of polynomials defined on the real axis and on the unit circle and to the Chebyshev procedure. A modern version of the Chebyshev recursion ((m)−T-recursion) is applied to check whether the Hankel and Toeplitz quadratic forms are positive definite, to determine the number of real (complex conjugate) roots of the polynomials, to localize the ordering of these roots, and to find bounds for the values of a function on a given set. We also consider the relation between the (m)−T-recursion and the method of moments in the study of Schrödinger operators for special classes of potentials.
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References
P. L. Chebyshev, “Sur l'interpolation par la methode des mondres carres,”Mem. Acad. Imper. Sci. St. Petersbourg,1, No. 15, 1–24 (1859).
J. C. Wheeler, “Modified moments and continuum fraction coefficients for the diatomic linear chain,”J. Chem. Phys,30, No. 1, 472–476 (1984).
W. Gautschi, “Orthogonal polynomials-constructive theory and applications,”J. Comput. Appl. Math.,12/13, 61–75 (1985).
M. Krein and M. Naimark, “The method of symmetric and Hermite quadratic forms in the theory of isolation of the roots of algebraic equations,”Linear Multilinear Algebra,10, 265–308 (1981).
I. E. Ovcharenko, “Scalar products in the space of polynomials and positivity,”Dokl. Akad. Nauk Ukr. SSR, Ser A, No. 7, 17–21 (1990).
I. E. Ovcharenko, “Applications of the Chebyshev recursion,”Dokl. Akad. Nauk SSSR,319, No. 2, 287–291 (1991).
R. A. Ugrinovskii, “On a trigonometric problem of moments with a gap; the Levinson algorithm and applications,”Ukr. Mat. Zh.,44, No. 5, 713–716.
S. A. Korzh, I. E. Ovcharenko, and R. A. Ugrinovskii, “Chebyshev's recursion and its applications,” in:Abstracts of XVI All-Union School on Theory of Operators in Function Spaces [in Russian], Nizhnii Novgorod (1991), p. 114.
N. I. Akhiezer,Classical Moment Problem [in Russian], Fizmatgiz, Moscow (1961).
N. Levinson, “The Wiener RMS (root-mean-square) error criterion in filter design and prediction,”J. Math. Phys.,25, 261–278 (1947).
A. I. Notik, A. I. Knafel', V. I. Turchin, et al., “Spectral analysis based on a continuum analog of the maximal entropy method,”Radiotekh. Elektron., No. 9, 1904–1912 (1990).
F. R. Gantmakher,Matrix Theory [in Russian], Nauka, Moscow (1988).
A. K. Sushkevich,The Fundamentals of Higher Algebra [in Russian], Gostekhizdat, Moscow (1941).
M. A. Kowalski, “Representation of scalar product on space of polynomials,”Acta Math. Hung.,46, No. 1–2, 102–109 (1985).
I. E. Ovcharenko, “Positive definiteness; orthogonal systems; positivity,”Vestn. Khark. Univ., Prikl. Mat. Mekh., 3–15 (1992).
C. R. Handy, D. Bessis, and T. D. Morley, “Generating quantum energy bounds by the moment method: A linear-programming approach,”Phys. Rev. A.,37, No. 12, 4557–4569 (1988).
C. R. Handy and D. Bessis, “Rapidly lower bounds for Schrödinger equation ground state energy,”Phys. Rev. Lett.,55, No. 9, 931–934 (1985).
C. R. Handy, “Moment method quantization of a linear differential equation for¦Ψ¦2,”Phys. Rev. A.,36, No. 9, 4411–4416 (1987).
E. R. Vrscay and C. R. Handy, “The perturbed two-dimensional oscillator: eigenvalues and infinite-fields limits via continued fractions renormalized theory and moment methods,”J. Phys. A: Math. Gen.,22, No. 2, 823–834 (1989).
C. R. Handy and J. Q. Pei, “Moment-method analysis of ground state of discretized bosonic systems,”Phys. Rev. A.,38, No. 7, 3175–3181 (1988).
H. S. Wall,Analytic Theory of Continued Fractions, Van Nostrand, New York (1948).
L. De Branges,Hilbert Spaces of Entire Functions, Prentice-Hall, Englewood Cliffs, N. J. (1968).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 5, pp. 626–646, May, 1993.
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Korzh, S.A., Ovcharenko, I.E. & Ugrinovskii, R.A. Chebyshev's recursion: Analytic principles and applications. Ukr Math J 45, 684–705 (1993). https://doi.org/10.1007/BF01058206
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DOI: https://doi.org/10.1007/BF01058206